
where the trace is taken over the extended Hilbert space containing non-singlet modes.
(The micro-canonical partition function can be obtained by constraining the energy.)
The operator \(\hat{g}\)  is the gauge transformation corresponding to \(g\in G\) ,
and \(\frac{1}{{\rm vol}(G)}\int _G dg \hat{g}\)  is the projector to the singlet sector.\(\endcsname \)The integral is taken over the Haar measure. When \(G\)  is discrete, a sum is taken instead of the integral.

Quantum mechanics of \(N\)  indistinguishable bosons is a special case of gauge theory,
whose gauge group S\(_N\)  is permutations of particles [1]}, [2]}.
In the free limit, the energy-einegstates are products of one-particle eigenstates, schematically expressed as
\(|\phi _1,\phi _2,\cdots ,\phi _N\rangle \) . A permutation \(g\in G={\rm S}_N\)  transforms such a state to \(|\phi _{g(1)},\phi _{g(2)},\cdots ,\phi _{g(N)}\rangle \) . Generic excited states are not invariant under any nontrivial permutation, and hence, only \(g=\textbf {1}\)  has a nonzero contribution to the partition function. On the other hand, the ground state is invariant under any permutation,
and there is an enhancement factor \(N!\)  that corresponds to \({\rm vol}(G)\) .
This enhancement triggers the Bose-Einstein condensation [3]}, and the Bose-Einstein condensate is characterized by the invariance under S\(_N\)  [2]}.
In the same manner, the confining ground state of \({\rm SU}(N)\)  gauge theory is invariant under \(G={\rm SU}(N)\) ,
and receives the enhancement factor \({\rm vol}(G)\)  that triggers the color confinement [1]}.\(\endcsname \)More precisely, \(G\)  is the set of all \({\rm SU}(N)\) -valued functions \(g(\vec{x})\) , where \(\vec{x}\)  is spatial coordinates in QFT.

